发表机构
Dipartimento di Matematica, Universitá degli Studi di Bari; INFN, Sezione di Bari; School of Physics, Trinity College Dublin; School of Physical and Mathematical Sciences, Nanyang Technological University; Dipartimento di Fisica, Università degli Studi di Bari（巴里大学数学系; 意大利国家核物理研究所巴里分部; 都柏林三一学院物理学院; 南洋理工大学物理与数学科学学院; 巴里大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探究随机矩阵半群的极限，发现其与保极限分布的半群不交换，针对不变分布为吉布斯向量的情况，刻画了极限半群的结构并引入块三角优超概念，分析了Rényi α-熵的相关行为。
AI 中文摘要
我们研究由不变分布定义的随机矩阵半群的极限。给定依赖于参数β的概率向量γ(β)，我们引入当β→∞时，对应保γ(β)的随机矩阵半群的收敛概念，并研究所得极限的结构。一般而言，极限半群与保极限分布的半群不同，表明这两种操作不交换。我们为这类极限半群建立了一般框架，并详细研究了不变分布为逆温度β下的吉布斯向量的情况。我们证明，极限半群由受额外次随机性约束的块上三角随机矩阵构成；我们刻画并枚举其极元，确定半群作用在概率向量上诱导的预序。所得的块三角优超概念介于普通优超与上三角（即无序）优超之间。我们证明其完全由有限个单调量刻画，并分析当β→∞时对应的Rényi α-熵的行为。
英文摘要
We investigate limits of semigroups of stochastic matrices defined by their invariant distribution. Given probability vectors $γ(β)$ depending on a parameter $β$, we introduce a notion of convergence as $β\to\infty$ for the corresponding semigroups of $γ(β)$-preserving stochastic matrices and investigate the structure of the resulting limit. In general, the limiting semigroup differs from the semigroup preserving the limiting distribution, showing that these two operations do not commute. We develop a general framework for such limiting semigroups and study in detail the case in which the invariant distributions are Gibbs vectors at the inverse temperature $β$. We show that the limiting semigroup consists of block-upper-triangular stochastic matrices subject to additional substochasticity constraints. We characterise and enumerate their extremal elements and determine the preorder on probability vectors induced by the action of the semigroup. The resulting notion of Block-Triangular majorisation interpolates between ordinary majorisation and upper triangular (aka unordered) majorisation. We show that it is completely characterised by a finite family of monotones and analyse the corresponding behaviour of Rényi $α$-entropies as $β\to\infty$.
Comments21 pages, 3 figure; Comments welcome!
